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The General Theory of Set Valued Semidynamical Systems

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Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 158)

Abstract

A set-valued mapping or correspondence, ө: rr is defined along with a family of correspondences, parameterized by t, in the following manner:
$$ {{\theta }^{0}}(X): = X $$
(3.1)
$$ {{\theta }^{t}}(X): = \theta ({{\theta }^{{t - 1}}}(X))\quad \forall t\varepsilon {{{\rm N}}_{ + }} $$
(3.2)
where x ∈r: Q × P. Hence Θt is the formal solution of the set valued difference equation Xt+1. Since Θ is a USC, Θt is a USC for all t ∈ N+ when Θ(X) =UX∈X Θ{X}. The family of correspondences {Θt|t ∈ N+ forms a semigroup of operators, for Θ0 is an identity operator, and for all r, s ∈ N+ and × ∈ r. Θrs(X}}=Θr+s. The value of Θt(X) is the set in which motion starting in the set X will be located after the duration of length t. This semigroup is analogous to the single valued solution of a difference or differential equation in single valued systems where, for each t, an initial condition corresponding to X is mapped to some other position in the state space. For such single valued systems, {Θt} is a set of continuous mappings.

Keywords

Accumulation Point Global Attractor Strong Attractor Closed Mapping Weak Attractor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1978

Authors and Affiliations

  1. 1.Modelling Research Group, Department of EconomicsUniversity of Southern CaliforniaUniversity Park, Los AngelesUSA

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